In this work we take a Category Theoretic perspective on the relationship between probabilistic modeling and function approximation. We begin by defining two extensions of function composition to stochastic process subordination: one based on the co-Kleisli category under the comonad (Omega x -) and one based on the parameterization of a category with a Lawvere theory. We show how these extensions relate to the category Stoch and other Markov Categories. Next, we apply the Para construction to extend stochastic processes to parameterized statistical models and we define a way to compose the likelihood functions of these models. We conclude with a demonstration of how the Maximum Likelihood Estimation procedure defines an identity-on-objects functor from the category of statistical models to the category of Learners. Code to accompany this paper can be found at https://github.com/dshieble/Categorical_Stochastic_Processes_and_Likelihood

翻译：在这项工作中,我们从分类理论角度看待概率模型和功能近似之间的关系。我们首先从界定功能构成的两个延伸扩展到随机过程从下从属关系(Omega x - ) :一个基于comonad (Omega x - ) 的共Kleisli 类别,另一个基于Lawvere 理论的类别参数化。我们展示这些扩展与Stoch 和其他Markov 类别的关系。接下来,我们应用 Para 构建扩展随机分析过程以参数化统计模型,我们定义了计算这些模型可能函数的方法。我们最后我们演示了最大相似性模拟程序如何从统计模型类别到学习者类别中确定身份-对对象的替代变量。本文所附的代码可以在 https://github.com/dshieble/Categorical_Stochicistic_Processes_and_Cellihood中找到。